Chapter 3 Probability Initial Objective: Develop procedures to determine the # of elements in a set without resorting to listing them first. 3-6 Counting

Order is important Worksheet: Coloring outfits to determine the total number of combinations possible. ions/Combinations-AS-ShortsandShirts.pdf

Possible Results Yellow shirt, Brown shorts Yellow shirt, Black shorts Yellow shirt, Green shorts Yellow shirt, Purple shorts Orange shirt, Brown shorts Orange shirt, Black shorts Orange shirt, Green shorts Orange shirt, Purple shorts Blue shirt, Brown shorts Blue shirt, Black shorts Blue shirt, Green shorts Blue shirt, Purple shorts Red shirt, Brown shorts Red shirt, Black shorts Red shirt, Green shorts Red shirt, Purple shorts

◦ Website Bobby Bear -Students can pick an outfit for Bobbie Bear and customize the outfit similar to the one on the activity sheet. =3 =3

Discussion Questions How did your prediction compare to your actual answer? How do you explain this? Which method would be more efficient for finding the total number of outfits: multiplying, drawing a tree diagram, or making a table? Which method would be more useful for identifying the different combinations (outfits) possible: multiplying, drawing a tree diagram, or making a table?

Fundamental / Basic Counting Rule For a sequence of 2 events in which the 1 st event can occur m ways and the 2 nd event can over n ways, the events together can occur a total of m x n ways. Example In designing a computer, if a byte is defined to be a sequence of 8 bits, and each bit must be a 0 or 1, how many different bytes are possible?

Answer Example In designing a computer, if a byte is defined to be a sequence of 8 bits, and each bit must be a 0 or 1, how many different bytes are possible? Answer: Since each bit can occur in 2 ways (0 or 1) and we have a sequence of 8 bits, the total # of different possibilities is given by 2x2x2x2x2x2x2x2= 256

Factorial Rule- n different items can be arranged in order n! different ways Ex. 5! = 5x4x3x2x1= 120 0! =1 Key is on your calculator Note: The factorial rule reflects that fact that the 1 st item maybe selected n different ways; the 2 nd item maybe selected n-1 ways, and so on.

Factorial Examples Example: How many possible ways routes are there to 3 different cities? Example: What about possible routes each of the 50 states?

What if you don’t what to include all of the items available? Order is still important!

Permutations Rule The # of permutations(or arrangements) of r items selected from n available items is n p r =n!/(n-r)! Note: rearrangements of the same items to be different. Look at page 155 for an example.

What if we tend to select r items from n available items, but do not take order into account? ◦ We are really concerned with possible combinations.

B: Order is not important When different ordering of the same items are to be counted separately, we have a permutation problem, but when different orderings are not to be counted separately, we have a combination problem.

Combinations Rule The # of combinations of r items selected from n available items is n C r = [Other notations for n C r are

Run through page 156 to 158 Could you image not having these counting techniques, it would take hours and hours to come up with all the possibilities.